Question: Simplify the following expression and state the condition under which the simplification is valid. $k = \dfrac{r^2 - 64}{r - 8}$
First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = r$ $ b = \sqrt{64} = -8$ So we can rewrite the expression as: $k = \dfrac{({r} {-8})({r} + {8})} {r - 8} $ We can divide the numerator and denominator by $(r - 8)$ on condition that $r \neq 8$ Therefore $k = r + 8; r \neq 8$